Retrospective Rating
October 03, 2015
- Recall: Prospective WC Rating
- Retrospective Rating: Intro
- Retrospective Rating: w/o Occurence Limit
- Understanding the Net Insurance Charge: w/o Net Insurance Charge
- Retrospective Rating: with Occurence Limit
- NCCI Filings for Optional Components
- Table M
- Limited Loss Table M
- Table L
- Table L vs NCCI
- Example of Table L, LLM, and M
- Using NCCI Manual
- Cliff’s Summary
- Types of Exam Questions
- Must Know Formulas
- Questions
Recall: Prospective WC Rating
2000 Q6
\(\begin{align} \text{Premium} = &\sum\limits_{i \in class} \dfrac{Payroll_i}{100} \text{Manual Rate}_i &\cdots (1)\\ &\times \: \text{Exp Mod} &\cdots (2)\\ &\times \: \text{Sch Mod} &\cdots (3)\\ &\times \: (1 - \text{Premium Discount %}) &\cdots (4) \end{align}\)
\((1)\) Manual Premium
\((1) \times (2)\) Modified Premium
\((1) \times (2) \times (3)\) Standard Premium
- Authorized rates + E-mod + Loss constant + Min Premium
\((1) \times (2) \times (3) \times (4)\) Guaranteed Cost Premium
- Assume no expense fee
- Reflects lower proportion of fixed expense for larger risk
- Usually calculated by Standard Premium tiers
Retrospective Rating: Intro
Uses insured’s loss experience during the current policy term to determine their current policy premium
Initial premium collected at the start (deposit premium) \(\Rightarrow\) Adjust based on reported loss starting 6 months after policy expires, then every 12 months after
\(\text{Premium} = \left[\text{U/W Expense & Profit xTax} + \left(1 + \dfrac{\mathrm{E}[LAE]}{\mathrm{E}[Loss]}\right) \times Loss_{actual}\right] \times \text{Tax Multiplier}\)
There’s always going to be a max and min premium or else this would defeat the purpose of risk transfer
Caps are applied to the premium amount \(\equiv\) Capping actual losses at max/min aggregate loss limits \(\Rightarrow\) Add a charge to the premium calculation to account for this
In addition to aggregate caps, can also limit losses per occurrence (Use limited losses in the retro premium calculation) \(\Rightarrow\) A separate charge to the premium calculation to account for the per occurrence limit
With both occurrence and agg limits, need to consider the overlap between the 2 charges
The starting point for retro premium is the standard premium
- GCP only uses expected losses to calculate the premium
- Retro premiums use a combination of expected loss (i.e., the insurance charge and savings) and actual loss to calculate the premiums
Retrospective Rating: w/o Occurence Limit
Important Formulas
NCCI retro premium formula for each individual policy
\(R = (b + C A + PCV) T\) Memorize
\(R\): Retro premium @ t \(\in [G, H]\)
\(b\): Basic premium = Profit + Charge for max/min premium + non-LAE expense xTax
- Charge for max/min premium \(\equiv\) Net insurance charge
\(A\): Reported loss @ t (may include ALAE)
\(C\): Loss conversion factor \(= 1 + \dfrac{\mathrm{E}[LAE]}{\mathrm{E}[Loss]}\)
- If \(A\) includes ALAE \(\Rightarrow\) \(= 1 + \dfrac{\mathrm{E}[ULAE]}{\mathrm{E}[Loss \& LAE]}\)
\(T\): Tax multiplier \(=\dfrac{0.2 + PLR(1+\mu)}{0.2 + PLR} \left({\dfrac{1}{1-\tau}}\right)\) Memorize
\(PLR = \text{Permissible Loss Ratio} = \dfrac{\mathrm{E}[A]}{P}\)
\(\mu = \text{Assessments}\)
\(\tau = \text{Premium Based Tax}\)
For an interstate risk, use a weighted average of the individual state tax multipliers with the standard premium for each state as the weights
\(V\): Retro development factor (Optional)
\(= (1-\frac{1}{CDF}) \Rightarrow \text{Unreported Losses}\)
\(C \times A + P \times C \times V \equiv \text{B-F Ultimate}\)
To stabilize premium adjustments
Only used for the first 3 adjustments
\(V\) are fixed at inception but they change over time according to the plan from the outset
Since losses develop over time, without a retrospective development factor, usually losses will be lower at early adjustments, which will cause refunds to the insured. Once losses develop higher, the insured will have to pay the money back. Using a retrospective development factor will reduce the back-and-forth payments, reduce the credit risk of collecting future premiums, and allow the insurer to have more premium at early adjustments to use in earning additional investment income.
\(P\): Standard Premium
For a balanced plan:
Expected retrospective premium = premium for a prospectively rated policy
\(b = e - (C - 1)\mathrm{E}[A] + CI\)
- \(b\): Insurer expense + loss control + premium audit G&A + Adj for limiting ratio + profit contingencies
- \(e\): Total expense xTax and profit
- \(I\): Net insurance charge for the max/min premium
- Since \(e\) includes all expenses including LAE (except taxes), we subtract off the expected LAE, since we will instead use the LAE related to actual losses from the \(CA\) in the retro premium equation
- Proof
2001 Q32
Understanding the Net Insurance Charge: w/o Net Insurance Charge
Ignore the retro development for now
\(R = (b + C \times A) \times T \: \: \: \text{for} \: H \leq R \leq G\)
Max and min premiums \(\equiv\) Aggregate limits on losses
Formula
2001 Q31
\(\begin{align} H &= (b + C \times A_H) \times T &= \text{min}\\ G &= (b + C \times A_G) \times T &= \text{max}\\ \end{align}\)
\(R = (b + C \times {\color{blue}L}) \times T \: \: \: \: {\color{blue}L} = \left\{ \begin{array} AA_H & A \leq A_H\\ A & A_H < A <A_G\\ A_G & A \geq A_G\\ \end{array} \right.\)
Understand Formula
\(\begin{array}{ccc} \text{Net Insurance Charge} = &I = (&\text{Ins Chg} &- &\text{Ins Saving}&) \times \mathrm{E}[A]\\ &I = (&\dfrac{\mathrm{E}[Loss > A_G]}{\mathrm{E}[A]} &- &\dfrac{\mathrm{E}[Loss < A_H]}{\mathrm{E}[A]}&) \times \mathrm{E}[A]\\ &I = (&\text{Tbl M Max Prem Chg} &- &\text{Tbl M Min Prem Savings}&) \times \mathrm{E}[A]\\ &I = (&\phi(r_G) &- &\psi(r_H)&) \times \mathrm{E}[A]\\ &I = (&\phi(\dfrac{A_G}{\mathrm{E}[A]}) &- &\psi(\dfrac{A_H}{\mathrm{E}[A]})&) \times \mathrm{E}[A]\\ \end{array}\)
Table M Terminology
Rows are entry ratios = \(r = \dfrac{Loss_{Actual}}{Loss_{Expected}}\) Understand
- \(\because\) Standardized \(\therefore\) Table M can be applied to various states and LOB
Columns are for different risk size groups
- Since distn of charges vary by risk size \(\Rightarrow\) Curve differ by risk size
Entry ratio at maximum premium \(= r_G = \dfrac{A_G}{\mathrm{E}[A]}\)
Entry ratio at minimum premium \(= r_H = \dfrac{A_H}{\mathrm{E}[A]}\)
Retrospective Rating: with Occurence Limit
Per occurrence limit impact the likelihood of hitting the max/min premiums \(\Rightarrow\) Table M not appropriate \(\Rightarrow\) Need to recognize the potential overlap between the occurrence and aggregate limits
3 options to handle this:
Option 1: Limited Loss Table M
Important Formulas
\(R^* = (b^{LLM} + CA^* + PCF + PCV) T\)Memorize
\(b^{LLM}\): From the limited loss Table M
\(A^*\): Reported limited loss @ t (might include ALAE)
- \(F = ELF = \dfrac{\mathrm{E}[Loss \geq \text{Per Occ Limit}]}{\text{Standard Premium}}\)
- Varies by State + HG + Occ Limit
- Recall Robertson paper
\(A \neq A^* + PF\)
\(\mathrm{E}[A] = \mathrm{E}[A^*] = PF\)
\(PCF\) and \(PCV\) are electives
\(V\): Development different with occ limit than w/o
Theoretically correct approach
Caveat:
Requires a large number of tables
Charge need to vary by occ limit, entry ratio, risk size group, HG, etc
Option 2: Table M with ICRLL Adjustment (NCCI)
ICRLL = Insurance Charge Reflecting Loss Limitation
Change the Table M column used to be the column appropriate for a larger size risk
- Per occurrence limit reduces the skewness of the entry ratio distn \(\Rightarrow\) Similarly for larger risk size
Caveat:
- Table M doesn’t vary by loss limit \(\Rightarrow\) Compromises a bit on accuracy
Option 3: Table L (WCIRB)
\(R^* = (b^L + CA^* + PCV) T\)
Accurately correct for overlap between the occurrence and aggregate charges similar to the LLM
Combines occurrence charge with the aggregate charge
- Whereas LLM’s occ charge has to be added separately
WCIRB uses different HG than NCCI
\(V\) varies by occ limit
- \(\uparrow\) limits \(\uparrow\) \(V\) \(\because\) \(\uparrow\) potential for development
NCCI Filings for Optional Components
NCCI do full rates or loss costs depending on the state
Full Rates:
Files \(ELF\) from option 1
Files \(V\), Retro Development Factors, from option 1
Loss Costs:
Files \(ELPPF\) (XS Loss Pure Premium Factors)
\(\begin{array}{cccc} ELF &= &ELPPF &\times &ELR\\ \dfrac{\mathrm{E}[XSLoss]}{\text{Standard Premium}} &= &\dfrac{\mathrm{E}[XSLoss]}{\mathrm{E}[Loss]} &\times &\dfrac{\mathrm{E}[Loss]}{\text{Standard Premium}}\\ \end{array}\)
\(ELR\) from company’s own estimate
Use \(ELAEPPF\) if per occ limit includes Loss & ALAE
Files Development Retrospective Pure Premium Factors
\(\begin{array}{cccc} \text{Retro Dev Factor} &= &\text{Retro Dev PP Factor} &\times &ELR\\ \dfrac{\mathrm{E}[\text{Unreported Limited Loss}]}{\text{Standard Premium}} &= &\dfrac{\mathrm{E}[\text{Unreported Limited Loss}]}{\mathrm{E}[Loss]} &\times &\dfrac{\mathrm{E}[Loss]}{\text{Standard Premium}}\\ \end{array}\)
XS Loss and Allocated Expense Pure Premium Factors
Table M
Constructing Table M
Aggregate distn across a large number of risks
- For losses, LR, entry ratios, etc
- Each data point is a single policy for a single term
Table M Charge
Revisit! This seems wrong…
\(\begin{align} \text{Table M Charge} &= \dfrac{\sum_{r_i > r} Loss_i}{\sum_i Loss_i}\\ &= \dfrac{\text{Total Losses }\forall \text{ Policies w/ Losses > Entry Ratio r}}{\text{Total Losses }\forall \text{ Policies}} \end{align}\)
\(\begin{align} \text{Table M Savings} &= \dfrac{\sum_{r_i < r} Loss_i}{\sum_i Loss_i}\\ &= \dfrac{\text{Total Losses }\forall \text{ Policies w/ Losses < Entry Ratio r}}{\text{Total Losses }\forall \text{ Policies}} \end{align}\)
2 Components to build Table M
- Sample of experience from a group of similarly sized risks
- Risk measure can be actual losses, LR, etc
- \(\mathrm{E}[\textit{Experience Measure}]\) from the sample
Normalize table if the sample average \(\neq\) assumed expected value:
- Ignore actual expected \(\Rightarrow\) Use sample avg \(\Rightarrow\) \(\phi(0) = 1\)
- Divide charge and entry ratio columns by \(\phi(0)\) to normalize
Constructing Table M Example
| Annual Claim Total ($) | # of Policies |
|---|---|
| 1000 | 8 |
| 1500 | 3 |
| 4000 | 2 |
| 10000 | 5 |
| 15000 | 2 |
| 40000 | 1 |
Each risk has $10,000 of standard premium
\(\text{Max Premium} \Leftrightarrow\) 80% LR w.r.t. Standard Premium
\(\text{Min Premium} \Leftrightarrow\) 20% LR w.r.t. Standard Premium
Loss @ Max Premium = 80% \(\times\) $10,000 = $8,000 = \(A_G\)
Loss @ Min Premium = 20% \(\times\) $10,000 = $2,000 = \(A_H\)
Total # of Policies = 21
Total Losses = $140,500
\(\mathrm{E}[A] = \text{Expected Loss per Policy} = \dfrac{\text{Total Losses}}{\text{Total # of Policies}} =\) $6,690.48 = Total area under curve
# Transformation of the data to plot
d1_ecdf <-
d1 %>%
rbind(c(0,0),.) %>%
mutate(`% Policies` = `# of Policies` / sum(`# of Policies`),
`Claim EWCDF` = cumsum(`% Policies`))
# Note the addition of 0 row| Annual Claim Total ($) | # of Policies | % Policies | Claim EWCDF |
|---|---|---|---|
| 0 | 0 | 0.0000000 | 0.0000000 |
| 1000 | 8 | 0.3809524 | 0.3809524 |
| 1500 | 3 | 0.1428571 | 0.5238095 |
| 4000 | 2 | 0.0952381 | 0.6190476 |
| 10000 | 5 | 0.2380952 | 0.8571429 |
| 15000 | 2 | 0.0952381 | 0.9523810 |
| 40000 | 1 | 0.0476190 | 1.0000000 |
Note that the steps are drawn in a “vh” manner starting from origin
Table M Charge in dollars = Area under curve and above $8,000 line \(A_G\)
- If divided by total area = Table M Charge
Table M Savings in dollars = Area under $8,000 line \(A_G\) and above curve
- If divided by total area = Table M Savings
Net Insurance Charge = Difference of the Table M Charge in dollars and Table M Savings in dollars
3 Ways to Calculate the Areas
- Horizontal Slices (Table M Method)
- Calculating the entry ratio first
- Vertical Slices
- Calculate area within limit and then subtract it from the expected loss
- = Limiting losses at the max/min
Horizontal Slices (Table M Method)
Calculating the entry ratio first
- Used to get the “height” of the segment
Starting from the top of the graph
Cumulate the horizontal length of each section from top to bottom
- The % risks above \(r_i\) piece
Advantages:
- Efficient when calculating multiple \(\phi{r_i}\)
- Faster when there are many risks
Table M Construction from Sampled Loss
1. Start with table of sampled loss amount and respective frequency
| Annual Claim Total ($) | # of Policies |
|---|---|
| 1000 | 8 |
| 1500 | 3 |
| 4000 | 2 |
| 10000 | 5 |
| 15000 | 2 |
| 40000 | 1 |
2. Add rows for 0 loss and entry ratio
tbl_m <-
d1 %>%
rbind(c(0,0), c(minlr * sp,0), c(maxlr *sp,0)) %>%
arrange(`Annual Claim Total ($)`) %>%
rename(`# risks` = `# of Policies`)
kable(tbl_m, align = 'l')| Annual Claim Total ($) | # risks |
|---|---|
| 0 | 0 |
| 1000 | 8 |
| 1500 | 3 |
| 2000 | 0 |
| 4000 | 2 |
| 8000 | 0 |
| 10000 | 5 |
| 15000 | 2 |
| 40000 | 1 |
3. Calculate Entry Ratio \(r\)
- \(\dfrac{Loss_{Actual}}{Loss_{Expected}}\)
tbl_m <-
tbl_m %>%
mutate(`Entry Ratio (r)` = `Annual Claim Total ($)` / weighted.mean(`Annual Claim Total ($)`, `# risks`))
kable(tbl_m, align = 'l')| Annual Claim Total ($) | # risks | Entry Ratio (r) |
|---|---|---|
| 0 | 0 | 0.0000000 |
| 1000 | 8 | 0.1494662 |
| 1500 | 3 | 0.2241993 |
| 2000 | 0 | 0.2989324 |
| 4000 | 2 | 0.5978648 |
| 8000 | 0 | 1.1957295 |
| 10000 | 5 | 1.4946619 |
| 15000 | 2 | 2.2419929 |
| 40000 | 1 | 5.9786477 |
4. Calculate # of risk above each row (Loss)
tbl_m <-
tbl_m %>%
mutate(`# risks above` = sum(`# risks`) - cumsum(`# risks`))
kable(tbl_m, align = 'l')| Annual Claim Total ($) | # risks | Entry Ratio (r) | # risks above |
|---|---|---|---|
| 0 | 0 | 0.0000000 | 21 |
| 1000 | 8 | 0.1494662 | 13 |
| 1500 | 3 | 0.2241993 | 10 |
| 2000 | 0 | 0.2989324 | 10 |
| 4000 | 2 | 0.5978648 | 8 |
| 8000 | 0 | 1.1957295 | 8 |
| 10000 | 5 | 1.4946619 | 3 |
| 15000 | 2 | 2.2419929 | 1 |
| 40000 | 1 | 5.9786477 | 0 |
5. Calculate % of risk above each row
tbl_m <-
tbl_m %>%
mutate(`% risks above` = `# risks above` / sum(`# risks`))
kable(tbl_m, align = 'l')| Annual Claim Total ($) | # risks | Entry Ratio (r) | # risks above | % risks above |
|---|---|---|---|---|
| 0 | 0 | 0.0000000 | 21 | 1.0000000 |
| 1000 | 8 | 0.1494662 | 13 | 0.6190476 |
| 1500 | 3 | 0.2241993 | 10 | 0.4761905 |
| 2000 | 0 | 0.2989324 | 10 | 0.4761905 |
| 4000 | 2 | 0.5978648 | 8 | 0.3809524 |
| 8000 | 0 | 1.1957295 | 8 | 0.3809524 |
| 10000 | 5 | 1.4946619 | 3 | 0.1428571 |
| 15000 | 2 | 2.2419929 | 1 | 0.0476190 |
| 40000 | 1 | 5.9786477 | 0 | 0.0000000 |
6. Calculate the Table M Charge \(\phi(r)\)
\(\phi(r) = 0\) for the largest value of r \(\Rightarrow\) No \(\mathrm{E}[L]\) above the largest loss
\(\phi(r_i) = \phi(r_{i+1}) + (r_{i+1} - r_i)(\text{ % risks above }r_i)\) Memorize Formula
Savings \(= \psi(r) = \phi(r) + r - 1\) Memorize Formula
Vertical Slices
Looking at groups of losses based on the size of loss
Probably start from the right but don’t seem to matter
- For Tbl M Charge: \(\text{ % of Policies hitting cap} \times \text{XS Loss Amount}\)
- For Tbl M Savings: $
Advantages:
- Natural, since that’s how the data is presented
- Easier to understand
- Faster if just looking at 1 \(r\)
Understanding Table M
Important Concepts
Note that the y axis is now the entry ratios and not losses like it was in the earlier example
- This is like looking at a normal CDF but rotated along the x=y line
Defn:
\(Y = \dfrac{A}{\mathrm{E}[A]}\)
\(F(y) = \text{CDF of } y\)
Area under curve:
\(\mathrm{E}[Y] = \dfrac{\mathrm{E}[A]}{\mathrm{E}[A]} = 1\)
- Recall when the y axis was loss amount, the area was \(\mathrm{E}[Y]\)
Table M Charge and Savings Formulas
\(\phi(r) = \int\limits_r^{\infty} (y - r) dF(y) = \int\limits_r^{\infty} (y - r) f(y)dy\)
\(\psi(r) = \int\limits_r^{\infty} (r - y) dF(y)\)
Key Properties - Tbl M Charge Important
\(\phi(0) = 1, \: \phi(\infty) = 0\)
\(\phi'(r) = -G(r) \leq 0\)
- \(G(r) = 1 - F(r)\)
\(\phi''(r) = f(r)\)
Key Properties - Tbl M Savings Important
\(\psi(0) = 0, \: \psi(\infty) = \infty\)
\(\psi(r)'(f) = F(r) \geq 0\)
\(\psi''(r) = f(r) = \phi''(r)\)
Note from figure above, the area under \(r = \psi(r) + (1 - \phi(r)) \Rightarrow \psi(r) = \phi(r) + r -1\) style=“color:red;background-color:yellow”>Memorize
\(\phi\) is monotonically decreasing function of premium size
Table M Charges and Premium Size
Large sample and large risks \(\Rightarrow\) \(\downarrow\) variance in Loss Ratio and Entry Ratio than large sample but small risks
\(\lim\limits_{\text{Premium Size}\rightarrow \infty} Var(y) = 0\)
- Curve will flatten out and look like all risks have the same amount of losses
\(\phi(r) = \begin{cases} 1 - r &r \leq 1\\ 0 &r > 1 \end{cases}\)
CDF of Entry Ratio vs Entry Ratio
From TIA: Think about how Table Ms are built: you get the loss ratios for a bunch of similar sized risks, calculate the average loss ratio from these risks, and then calculate entry ratios for each risk as actual loss ratio / average loss ratio. The variance in the entry ratios is really just the variance in the actual loss ratios of the risks used to built the Table M. If the variance of these loss ratios goes to 0, then that is equivalent to saying all risks have the same loss ratio (aka, same entry ratio, aka same dollar amount of aggregate losses given equal premium sizes). So that is why the curve will flatten.
On the other hand, as \(\lim\limits_{\text{Premium Size} \rightarrow 0} \phi(r) = 1\)
Graphs of Entry Ratios vs Table M Charge/Savings as premium size increase
- Higher ELG (a different column) in Table M, which would have lower charges for a given entry ratio
Net Insurance Charge - w/o Occurence Limit
Important Concepts
\(R = (b + C \times {\color{blue}L}) \times T \: \: \: \: {\color{blue}L} = \left\{ \begin{array} AA_H & A \leq A_H\\ A & A_H < A <A_G\\ A_G & A \geq A_G\\ \end{array} \right.\)
\(A_H = r_H \mathrm{E}[A]\) and similarly for \(A_G\)
Blue area = \(\dfrac{\mathrm{E}[{\color{blue}L}]}{\mathrm{E}[A]}\)
- Recall that \(\mathrm{E}[{\color{blue}L}]\) is the expected limited loss
- = Expected Retro Premium? 2003 Q29
Memorize Formula
\(\begin{array}{ll} \dfrac{\mathrm{E}[L]}{\mathrm{E}[A]} &= 1 + \psi(r_H) - \phi(r_G) \\ \mathrm{E}[L] &= \mathrm{E}[A] + [\psi(r_H) - \phi(r_G)]\mathrm{E}[A] \\ \mathrm{E}[L] &= \mathrm{E}[A] - I \end{array}\)
- Recall: \(I = \mathrm{E}[A][\phi(r_G) - \psi(r_H)]\)
- Expected loss given max/min = Expected Loss - Net Insurance Charge
Memorize Understand
\(\begin{array}{lll} R &= (b + CL)T\\ \mathrm{E}[R] &= (b + C\mathrm{E}[L])T\\ \mathrm{E}[R] &= \big\{b + C(\mathrm{E}[A] - I)\big\}T & (1)\\ \end{array}\)
Recall if the plan is balance \(\Rightarrow\) \(\mathrm{E}[R] = \text{Guaranteed Cost Premium}\)
\(\begin{array}{lll} \text{Guaranteed Cost Premium} &= (e + \mathrm{E}[A])T & (2)\\ &=(1-D)P & (3)\\ \end{array}\)
- \(e\) = Total expense x tax & profit
- \(D\) = Premium discount \(= \sum (\text{Premium in Range}_i)(\text{Discount}_i)\)
- \(P\) = Standard Premiums
Combine \((1)\) and \((2)\)
\(\big\{b + C(\mathrm{E}[A] - I)\big\}T = (e + \mathrm{E}[A])T\)
\(b = e - (C - 1)\mathrm{E}[A] + CI\) Memorize Formula
- \(e - (C - 1)\mathrm{E}[A]\) is never capped
- \(CI\) is where the cap comes in
- Recall: Check \(e = \dfrac{1-D}{T} - PLR\)
\(T = \dfrac{0.2 + PLR(1+\mu)}{0.2 + PLR}\left(\dfrac{1}{1-\tau}\right)\)
2009 Q31, 2011 Q8
Question that ask for G or H usually use balance equation
Alt formula: \(b = (1 - D)\dfrac{P}{T} - C\mathrm{E}[A] + CI\)
Use this if we don’t’ have max/min premium
Table M Balance Equations
Basic premium and max/min premium depends on each other \(\Rightarrow\) Need trial and error to get the right Table M row
First balance equation
Value (change) difference
Retro premium formula with max loss:
\(H = (b + CA_H)T = (b + Cr_H\mathrm{E}[A])T\)
Start with the GCP formula and subtract \(H\)
\(\begin{array}{ll} \text{GCP} - {\color{blue}H} &\\ (e + \mathrm{E}[A])T - {\color{blue}H} &= \big\{b + C(\mathrm{E}[A] - I)\big\}T - {\color{blue}{(b + Cr_H\mathrm{E}[A])T}}\\ &= CT\big\{\mathrm{E}[A] - I - r_H\mathrm{E}[A]\big\}\\ &= CT\Big\{\mathrm{E}[A] - \mathrm{E}[A]\big\{\phi(r_G) - \psi(r_H)\big\} - r_H \mathrm{E}[A]\Big\}\\ &= C \: T \: \mathrm{E}[A] \big[ \psi(r_H) - r_H + 1 - \phi(r_G)\big] \\ \end{array}\)
Move everything to one side and we get: Important Memorize
\(\phi(r_H) - \phi(r_G) = \dfrac{(e + \mathrm{E}[A])T - H}{C\mathrm{E}[A]T} = \dfrac{(1-D)P-H}{C\mathrm{E}[A]T}\)
- Interpretation: Charge difference = to the difference between the expected retro premium E[R] and the minimum premium H, except for the factor CE[A]T
- 2003 Q29
Second balance equation
Entry difference, based \(G - H\)
Important Memorize
\(r_G - r_H = \dfrac{G - H}{C\mathrm{E}[A]T}\)
Can do all the above looking at them as a % of Standard Premium
Table M Search
- Use balance equations to get the difference and then look up the table
- Look for the Entry difference first then charge difference
Note:
- Charge difference \(\downarrow\) as you \(\uparrow\) entry ratios
- Entry ratio pair \(\Delta\) the charge difference by about 0.01
Limited Loss Table M
Constructing a Limited Loss Table M
Same as Table M but with limited losses \(A^*\) for each policy
Entry Ratio: \(r^* = \dfrac{A^*}{\mathrm{E}[A^*]}\)
Note that the denominator of the Entry Ratio is the \(\mathrm{E}[A^*]\), where as in Table L it’s not limited \(\mathrm{E}[A^*]\)
Understanding a Limited Loss Table M
Note that the y axis is now limited entry ratios \(y^*\)
Entry Ratio: \(Y^* = \dfrac{A^*}{\mathrm{E}[A^*]}\)
\(F^*(Y^*) = \text{CDF of }Y^*\)
Loss Elimination Ratio: \(k = LER = 1 - \dfrac{\mathrm{E}[A^*]}{\mathrm{E}[A]}\) Know
- \(\dfrac{\mathrm{E}[A^*]}{P} = \text{Unlimited ELR} - LER\)
Note that \(\mathrm{E}[Y^*] = \dfrac{\mathrm{E}[A^*]}{\mathrm{E}[A^*]}\) no surprise
But \(\mathrm{E}[Y] = \dfrac{\mathrm{E}[A]}{\mathrm{E}[A^*]} = 1 + k\dfrac{\mathrm{E}[A^*]}{\mathrm{E}[A]}\)
Integration and key properties all the same as Table M but with limited loss
Same thing for what happens when \(\lim\limits_{\text{Premium Size} \rightarrow \infty}\) and \(\lim\limits_{\text{Premium Size} \rightarrow 0}\)
Net Insurance Charge - LLM
\(R^* = (b^{LLM} + C{\color{blue}{L^*}}+PCF) T \: \: \: \: {\color{blue}{L^*}} = \left\{ \begin{array} AA_H & A^* \leq A_H\\ A^* & A_H < A^* <A_G\\ A_G & A^* \geq A_G\\ \end{array} \right.\)
\(+ PCV\) if to stabilize
Everything similar to Table M but now that the entry ratios \(r^*\) are limited and has some limied loss \(\mathrm{E}[A^*]\)
\(\mathrm{E}[L^*] = \mathrm{E}[A^*] - I^{LLM}\)
\(b^{LLM} = e - (C - 1)\mathrm{E}[A] + CI^{LLM}\)
- \(e - (C - 1)\mathrm{E}[A]\) is never capped
\(\phi^{LLM}(r_H^*) - \phi^{LLM}(r_G^*) = \dfrac{(e + \mathrm{E}[A])T - H}{C\mathrm{E}[A^*]T} = ? \dfrac{(1-D)P-H}{C\mathrm{E}[A^*]T}\)
\(r_G^* - r_H^* = \dfrac{G - H}{C\mathrm{E}[A^*]T}\)
Table L
Constructing Table L
Table L charge includes the charge for per occurrence limit
- LLM adds the per occ charge via ELF separately (by State + HG + occ limit)
Implicit charge for the occ limit \(\in \{0, k\}\)
- \(k\) is the Loss Elimination Ratio
- Incremental charge above the normal Table M charge.
- Varies by: occurrence limit, ELR, premium size, entry ratios
- Not vary by premium size would result in overcharging small risks and undercharging large risk
Table L uses different premium size groups and occurrence limits
The LER was derived from the combination of all premium groups
- That’s why entry ratio are based on expected unlimited losses \(\mathrm{E}[A]\)
CA version of Table M shows a higher charge than the CW average due to higher variation in the CA rates
\(\phi^L(r) - \phi(r)\) shows in the data was smaller than the LER implies due to overlap between occ limit and agg limit
Main Difference from the M and LLM
Entry ratios = \(\dfrac{A^*}{\mathrm{E}[A]} = \dfrac{\text{Actual Limited Loss}}{\text{Expected }\textbf{Unlimited}\text{ Loss}}\)
2013 Q15b
If Standard Premium \(\neq\) \(\forall\) risk \(\Rightarrow\) Create Tbl L with P instead of # of risk
- Denominator of Entry Ratios \(= \dfrac{\mathrm{E}[A]}{P}\)
- Numerator of Entry Ratios \(= \dfrac{A^*}{P}\)
- \(\phi^L(r_i) = \phi^L(r_{i+1}) + (r_{i+1} - r_i)(\text{ % Premium above }r_i)\)
2008 Q32
Understanding Table L
}Note that the y axis is \(Y = \dfrac{A^*}{\mathrm{E}[A]}\) and the CDF of interest is \(F^*(Y)\). However, the x axis of the graph is of \(F(Y)\)
- Axis = Original Table M since denominator of the Entry Ratio \(= \mathrm{E}[A]\)
Loss Elimination Ratio: \(k = LER = 1 - \dfrac{\mathrm{E}[A^*]}{\mathrm{E}[A]}\)
- Same as LLM
Area under \(F(Y) = 1\) same as Table M \(\Rightarrow\) Area under \(F^*(Y) = \mathrm{E}[Y] = \dfrac{\mathrm{E}[A^*]}{\mathrm{E}[A]} = 1 - k\)
- Based on graph
\(\phi^L(r) = \int\limits_r^{\infty} (y-r)dF^*(y) + k\) Remember the k
- Note the addition of k, charge for per occ limit, is in the table L charge
\(\psi^L(r) = \int\limits_0^{r} (r-y)dF^*(y)\)
Key properties all the same except:
\(\phi^L(\infty) = k\)
Net Insurance Charge - Table L
\(R^* = (b^L + C{\color{blue}{L^*}}) T \: \: \: \: {\color{blue}{L^*}} = \left\{ \begin{array} AA_H & A^* \leq A_H\\ A^* & A_H < A^* <A_G\\ A_G & A^* \geq A_G\\ \end{array} \right.\)
Everything same as Table M but now that we use \(I^L\)
\(\mathrm{E}[L^*] = \mathrm{E}[A^*] - I^{L}\)
\(b^{L} = e - (C - 1)\mathrm{E}[A] + CI^{L}\) * \(e - (C - 1)\mathrm{E}[A]\) is never capped
\(\phi^{L}(r_H) - \phi^{L}(r_G) = \dfrac{(e + \mathrm{E}[A])T - H}{C\mathrm{E}[A]T} = ? \dfrac{(1-D)P-H}{C\mathrm{E}[A]T}\)
\(r_G - r_H = \dfrac{G - H}{C\mathrm{E}[A]T}\)
2001 Q34
Table L vs NCCI
NCCI uses the ICRLL to \(\approx\) LLM
Table L Advantages:
- Mathematically accurate in how it accounts for the overlap
- No need for separate per occ charge (\(PCF\) in LLM)
- CA focus
Table L Disadvantages:
- Can’t be used for alternate loss limits
- Charge for a fixed limit is built into the table
- Requires large numbers of tables for CW
- Varies by Entry Ratio + State + Limit + Premium Size Group
- Likely by HG too since ELFs varies by HG
- Need regular update for inflation
- Aggregate distn \(\Delta\) shape
- With Table M you just update the \(\mathrm{E}[L]\)
- CW data might be more credible
Example of Table L, LLM, and M
See Manual for example
Using NCCI Manual
Some of the important sections below. Cancellation provision is not on the syllabus
Expected Loss Group Tables (Pt 4-B)
Expected Loss Group = Risk Size Groups
To determine the ELG:
- Adjusted Expected Losses \(= \mathrm{E}[A] \times \text{State HG Differential} \times \dfrac{1 + 0.8k}{1-k}\)
- \(k = LER\)
- \(\dfrac{1 + 0.8k}{1-k}\) is the ICRLL procedure
- LLM \(\approx\) Table M by adjusting the Adjusted Expected Losses to be for a ELG
- Use LLM balance equation instead if using this
- Look up ELG with the Adjusted Expected Losses
- Match the policy effective date with table effective date
\(\mathrm{E}[L]\) increase over time due to inflation. The ELG tables are updated so the curves so NCCI doesn’t have to update the curve. They’ll just push risk to different ELG.
Estimate portion of risk moving from ELG X to Y: \(\dfrac{\text{Portion of ELG X in new ELG Y}}{\text{Size of old ELG X}}\)
Since there is more variance in the loss ratio distn for smaller insureds than for larger insureds, there will be more variance in the entry ratio distn as well. As such, the tables in the NCCI manual reflect different columns for different insured sizes. Using a single distn for all insureds would result in overcharging the large insureds and undercharging the small insureds
Table M (Pt 4-C)
Each column is for different ELG
For low entry ratios, Table M savings are shown as well
Expense Ratio Tables (Pt 4-D)
Not tested since 2000
This is for the \(e\) in \(b = e - (C - 1)\mathrm{E}[A] + CI\)
Type A = Stock; Type B = non-Stock
Shows ELR, premium discount ranges and tax multipliers
AK Special Rating Values
2nd last page
- Items to get Adjusted Expected Losses
- HG Differentials; varies by HG
- ELPPF by occ limit + HG
- ELPPFs = LER \(k\)
Last page
- ELAEPPFs
- Retrospective PP Dev Factor
- By occ limit and adjustments
- Can convert into retrospective development factor \(V\)
Cliff’s Summary
Lee 2
Know how the entry ratio, charges and other relationships are represented as areas in the graph
Different forms of the charge and properties
Gillam & Snader 2
Using NCCI Table M
- Need Adj Expected Loss to get ELG to get know the column of Tbl M
Remember the basic premium formula only works if plan is balanced
- \(b = e - (C - 1)\mathrm{E}[A] + CI\)
Know how to build Table M
Know how to plug and play with all the equations
Brosius
Vertical and horizontal calculations of the tables
- Remember how to normalize
Table construction
Skurnick
Table L rationale pros cons
ICRLL rationale pros cons
Unbalanced retro calculation
Insurance charge and savings for Table L
NCCI 2
How to use the NCCI pages
- ELG, balance equations
- Use the table for the right year
- Look for entry difference first then charge
Types of Exam Questions
Lee 2
Interpret entry ratio plots
- 2000 Q19 (Interpret)
- 2001 Q30 (Explain its constructions)
- Understand that Table M is constructed horizontally
How charges are represented on graphs
- 2002 Q17
- 2003 Q8
- 2003 Q29 (interpret charge difference)
- ?Question?
- 2007 Q31 (both L and M)
Draw graphs and calculate charges
- 2005 Q33 (uniform)
- 2006 Q29 \(\star\)
- 2008 Q29 (uniform)
- 2009 Q9 \(\star\) (Calc expected retro premium)
- 2010 Q25
- 2012 Q21
- Convert \(r\) to LR to do this?
Basic premium from formula
- 2012 Q23 \(\star\)
Impact of change on losses or size
- 2012 Q21 b TIA
- 2013 Q15 b,c \(\star\)
Other
- 2002 Q14 (XS PP formula)
- Why is the denominator 1??Question?
Gillam & Snader 2
Using Table M
- 2000 Q5 (ELG)
Plug and play for \(b\), \(R\), \(GCP\), etc
- 2000 Q6 (Know how to do premium discount)
- 2000 Q9 (\(GCP\) and \(A\))
- 2001 Q8 (\(GCP\))
- 2001 Q31 (\(b\) formula and ELG)
- Not sure about part b
- TIA
- 2001 Q33 (Retro @18mo) \(\star\)
- 2002 Q15 (Charge difference)
- 2002 Q16 (net converted charge)
- 2002 Q40a (Retro @18mo)
- 2003 Q7 (Tax multiplier)
- 2003 Q10 (\(b\))
- 2004 Q18 (Retro @18mo)
- 2004 Q20 (Back out \(e\) with GCP and retro)
- 2005 Q30a (Tax multiplier)
- 2005 Q30b (Tax multiplier w/ interstate risk)
- 2005 Q32a (GCP and Retro Prem)
- 2006 Q30 \(\star\) (Retro, hit cap)
- 2006 Q32 \(\star\) (max prem factor, clever)
- 2010 Q27 \(\star\) (diff between limited and unlimited)
- 2010 Q29 \(\star\) (Multiple items)
- 2011 Q20 \(\star\) (GCP, expected loss is tricky)
- 2011 Q25 \(\star\) (max prem, need to 2 balance equation)
- 2012 Q19 \(\star\) (Retro @ 2nd adjustment)
- 2013 Q14 \(\star\) (expected retro, need that E[L]/E[A] formula)
- 2014 Q18 \(\star\)\(\star\)
Formula Manipulation
- 2001 Q32a \(\star\)
Build Table M
- 2000 Q48
Concepts
- 2001 Q10 (relationship between max/min/b)
- Do we need the assumed balance condition here?
- 2001 Q32b (iterative tbl M lookup)
- 2002 Q40b \(\star\) (Improve insured cashflow)
- 2003 Q33 \(\star\) (\(\Delta\) in exposure impacts)
- Not sure about b
- 2004 Q47 \(\star\) (How to lower b)
- 2005 Q32a (GCP and Retro Prem difference)
Tricky
- 2007 Q32 (defective)
- 2007 Q35 (Balanced or not)
- 2009 Q20 (defective)
Brosius
Insurance charge/savings
- 2002 Q36 (vertical)
- 2004 Q43 (vertical, tricky arithmetics)
- 2005 Q8 (vertical, charge/savings relationship)
- 2006 Q9 (table construction)
- 2007 Q30 (vertical)
- 2007 Q34 (table construction)
- 2008 Q28b (table construction)
- 2013 Q12 (table construction)
- 2014 Q13 \(\star\) (Backout numbers)
- Practice \(\star\) (table construction with normalize)
Concepts
- 2008 Q28a (vertical vs horizontal)
Skurnick
Understandig Table L and ICRLL
- 2000 Q45 \(\star\) (Tbl L and ICRLL, Pros Cons)
- 2002 Q38
- 2006 Q34 \(\star\) (per occ charge)
- 2008 Q35 (same as 2000 Q45)
- 2010 Q31 \(\star\) (ICRLL proc)
- 2011 Q22 bc
- 2013 Q13 \(\star\) (Relationship with prem size and properties)
- 2014 Q12 a \(\star\) (build table from graph)
- 2014 Q12 b \(\star\) (Explain savings)
Unbalanced retro calc
- 2001 Q34 \(\star\) (Unbalance and then rebalance)
- Good questions to understand
Table Construction
- 2003 Q30 (L and M)
- 2009 Q32 (L and M)
- 2011 Q22a (L)
- 2012 Q18 \(\star\)
Calculate insurance charge/savings
- 2004 Q44 (Vertical, L and M)
- 2006 Q7 (Vertical, L)
- Why is the per-occ charge not always just the LER??Question?
- 2008 Q32 \(\star\star\) (Table L with different SP risk)
- Use dollar amount or Tbl L with premium (instead of risk)
- Should understand this
Must Know Formulas
NCCI 2
Plug and play
- 2001 Q9 (LLM charge difference)
- 2002 Q35 \(\star\)
- expense and contingency in \(b\) = w/o the \(CI\)
- \(CI\) (Get ELG with LER + balance equation \(\rightarrow\) tbl M search)
- Total
- 2003 Q31 (ELG and balance equations and D)
- 2004 Q45 (similar to 2002 Q35)
- 2005 Q32 (Just charge)
- 2006 Q35 \(\star\) (backout ELF)
- 2008 Q36 (similar to 2002 Q35)
- 2009 Q30 (similar to 2002 Q35)
- 2011 Q21 \(\star\) (Alaska ELG)
- Lookup HG differential and ELAEPPF
- 2014 Q15 \(\star\) (inflation impact)
- 2014 Q17 \(\star\)
- GCP (Tax is applied to final P)
- \(b^{LLM}\)
- Impact of higher per occ limit
Concepts
- 2003 Q32 (Why varies by ELG)
Table M
Net Charge (For LLM use \(\mathrm{E}[A^*]\) and LLM charges):
- \(I = \mathrm{E}[A][\phi(r_G) - \psi(r_H)]\)
Savings:
- \(\psi(r) = \phi(r) + r -1\)
Other Relationships:
- \(\begin{array}{ll} \mathrm{E}[L] &= \mathrm{E}[A] \left\{1 + \psi(r_H) - \phi(r_G)\right\} \\ &= \mathrm{E}[A] - I \\ \end{array}\)
Table construction:
- \(\phi(r_i) = \phi(r_{i+1}) + (r_{i+1} - r_i)(\text{ % risks above }r_i)\)
Reading Table M
Balance Equations (Use \(\mathrm{E}[A^*]\) for the denominator if there’s occ limit):
\(\phi(r_H) - \phi(r_G) = \dfrac{(e + \mathrm{E}[A])T - H}{C\mathrm{E}[A]T}\)
\(r_G - r_H = \dfrac{G - H}{C\mathrm{E}[A]T}\)
Adjusted Expected Losses:
- \(= \mathrm{E}[A] \times \text{State HG Differential} \times \dfrac{1 + 0.8k}{1-k}\)
Questions
What is the x-axis for the Table LLM
What is the state differential in finding the Adjusted Expected Loss
What is that E[L]/E[A] formula
About This
Just a bunch of notes for Exam 8